Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,10,8)(7,11,9,12)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(29,30,34,35)(31,32,37,36)(33,38,41,42)(39,40,44,43);(5,7,10,9)(6,11,8,12)(15,22)(18,21)(23,28)(26,27)(30,35)(31,37)(33,41)(40,43);(6,8)(11,12)(13,15)(14,18)(16,23)(17,22)(19,26)(20,21)(24,27)(25,28)(29,31,34,37)(30,36,35,32)(33,39,41,44)(38,43,42,40);(1,2)(3,4)(5,8,10,6)(7,12,9,11)(13,14)(15,21)(16,19)(17,20)(18,22)(23,27)(24,25)(26,28)(29,32,34,36)(30,37,35,31)(33,40,41,43)(38,44,42,39);(5,9,10,7)(6,12,8,11)(13,16,17,25)(14,19,20,24)(15,23,22,28)(18,26,21,27)(29,33)(30,38)(31,39)(32,40)(34,41)(35,42)(36,43)(37,44)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=f,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4842812);